Topology of Algebraic Varieties (Spring 2016)


Historical background
Algebraic geometry is concerned with the relation between algebra and geometric objects, whereas algebraic topology deals with attaching invariants of an algebraic nature to topological spaces. But the path from the former to the latter (which is the topic of this course) is a hidden one and anything but straightforward. 
   The first result of substance of this type, obtained in the 19th century, says that a homogenenous polynomial F in three complex variables of degree d  whose partial derivatives have as common zero set only the origin (0,0,0) defines a nonsingular curve in the complex projective plane whose underlying topological space is a surface of genus (d-1)(d-2)/2. Further progress of this type in complex dimension 2 was made by Picard & Simart (1897), but it was Lefschetz who uncovered in his monograph L'Analysis situs et la géométrie algébrique (1924) the nature of its higher dimensional generalization, and indeed,  much of the rich structure of the algebraic topology of general complex projective manifolds. The tools and technique developed by him are still in use,  among them the notion of a Lefschetz pencil that is used to prove the Lefschetz hyperplane theorem.  Still one the most important results (known as the Hard Lefschetz theorem) received a complete proof only through the work of Hodge (1944). As this proof was based on Hodge’s theory of harmonic forms, it represented  highly nontrivial input from analysis and complex differential geometry. After Weil had formulated his famous conjectures in 1949, there was a sudden urgency to extend these results to varieties over an arbitrily algebraically closed field. This was what perhaps Grothendieck drove most in his rebuilding of the foundations of algebraic geometry. This enterprise was remarkably successful: Grothendieck’s many major contributions (often obtained in collaboration with his disciples) came close to completing this project and it was his former student  Deligne, who eventually not only proved the last of the Weil’s conjectures (1973), but also gave a proof of the hard Lefschetz theorem in this setting. Both Grothendieck (1966) and Deligne (1978) were awarded a Fields medal for their work. 
   There have been also developments regarding arbitrary (possibly singular and noncompact) varieties. Fulton-Lazarsfeld and Deligne have shown that for noncompact projective manifolds some form of the Lefschetz theorems still subsist. Goresky-MacPherson developed Intersection Cohomology (1977) as an intermediate between homology and cohomology for the purpose of retaining Alexander duality for singular spaces (for manifolds this is just ordinary cohomology). This cohomology turned out to be tailormade for generalizing the  Lefschetz and Hodge picture to projective manifolds. Indeed,  it was subsequently shown (with the most important contributions by  Goresky-MacPherson, Beilinson-Bernstein-Deligne-Gabber, Morihiko Saito) that the main theorems then  continue to hold. But this also gave rise to  sheaves  that were new and unexpected. An example is the Decomposition Theorem, a property about morphisms of varieties, rather than of a single variety, which has had numerous applications. Another example is the most natural form of the so-called Riemann-Hilbert correspondence (a correspondence on a complex manifold  between constructible  sheaves and D-modules, or rather between their derived categories).

Course syllabus
The dual of a projective variety and Lefschetz pencils 
The weak and hard Lefschetz theorems
Picard-Lefschetz formulae and global monodromy groups
The Leray spectral sequence
Stratified spaces, Lefschetz theory for singular varieties
Intersection cohomology, weak Lefschetz for 
intersection cohomology
Constructible sheaves, its derived category and its stability properties, perverse sheaves
Quasi-unipotence and the invariant cycle theorem
D-modules and mixed Hodge modules (overview)

Prerequisites
For most of the course, basic knowledge of algebraic geometry (such as in AG I)  and Algebraic Topology. Towards the end, when more advanced material is needed, we make up for deficiencies as we proceed.

Literature (will be expanded)
A set of notes for this course  is posted in installments here, although these will probably lag behind on the oral lectures.
Here is a somewhat 
artificial division by theme.

Lefschetz theory (classical and semi-classical):
K. Lamotke: The topology of complex varieties after S. Lefschetz, Topology 20, 15-51 (1986).
SGA 7-II: Groupes de Monodromie en Géométrie Algébrique (P. Deligne & N. Katz eds.), Springer LN in Math. 340 (1973).
C.H. Clemens: Degenerations of Kaehler manifolds, Duke Math. J. 22, 211-319 (1977)
W. Fulton and R. Lazarsfeld: Connectivity and its applications in algebraic geometry, in: Algebraic geometry (Chicago, Ill., 1980), 26–92, 
Springer LN in Math. 862 (1981).  
M. Goresky and R. MacPherson:
Stratified Morse Theory, Ergebnisse 14, Springer (1987).
P. Deligne: Théorème de Lefschetz et critères de dégénerescence de suites spectrales, Publ. Math. Sc. de l’IHES 35, 107-127 (1986).

Constructible sheaves and intersection cohomology (without Hodge theory)
A. Dimca: Sheaves in Topology, Universitext, Springer (2004).
B. Iversen: Cohomology of sheaves, Universitext, Springer (1986).
M. Kasihiwara: Quasi-unipotent constructible sheaves, J. Fac. Sci. Tokyo IA Math. 28, 757-773 (1982).
A. Borel et al.: Intersection cohomology, Progress in Math. 50, Birkhauser (1984).
A.A. Beilinson, J. Bernstein, P. Deligne: Faisceaux pervers, Astérisque 100 (1982).

Modern Hodge theory
P. Deligne: Theorie de Hodge I, II, III, 
   Actes Congres Intern. Math. Nice (1970), 425-430 (an outline of the underlying way of thinking),
   Publ. Math. de l’IHES 40, 5-75 (1971), 
Publ. Math. de l’IHES 44, 5-77 (1974).
P. Deligne:
Poids dans la Cohomologie des Varietés Algébriques, ICM Vancouver (1974), 79-85.
Topics in transcendental algebraic geometry (Ph. Griffiths ed.), Ann. of Math. Studies 106 (1984).
M. Saito: Introduction to mixed Hodge modules, in 
Astérisque 179/180, 145-162 (1987)

Place and time
Tuesdays (13:00-14:50) and Fridays (13:00-14:50),  starting March 1, 2016, in Room 1 of the Jin Chun Yuan West Building. 

Material covered
  March 1: Dual of a projective variety, Lefschetz pencils up to the proof of Prop. 1.10.
  March 4: Proof of 1.10, review CW complex, weak Lefschetz:
 formulation and first applications until Lemma 3.4.
  March 8: Finished proof of the weak Lefschetz theorem.
March 11: Monodromy, variation, Picard-Lefschetz formula, until Prop. 4.3.
March 15: Fundamental group of a hypersurface complement, invariant homology, vanishing lattices, until Section 5.
March 18: Around the Hard Lefschetz theorem. Concluded with the Lefschetz decomposition 5.9.
March 22: Finished Section 5, definition of quasi-unipotence and Zeta function.
March 25: Discussion of real oriented blowups.
March 29: Logmodel of the geometric monodromy, most of the proof of Prop. 6.2.
    April 1: Completion of proof of 6.2, stratified spaces, 
constructible sheaves (Section 8).
    April 5: Review homological algebra, adjoint pairs in algebraic topology, truncation functors 
(Section 7).
    April 8: Whitney stratifications, Thom isotopy, 
duality on a manifold. 
  April 12: Construction of the intersection cohomology complex. 
  April 15: Intersection cohomology complex in the analytic setting. Sample computations (finished Section 9). 
  April 19: Relative IH, local stratified model of a singularity, 
variation homomorphism revisited (Prop. 10.1).
  April 22: Weak Lefschetz theorem for constructible sheaves (up to Cor. 10.10).
  April 26: No class
  April 30: No class
     May 3: 
Holiday     
     May 6: Sheaf cohomology as a module over the cohomology ring, Tate twists, factorisation of the Lefschetz operator.
   May 10: Hard Lefschetz and primitive decompostion for IH (Section 11), derived category, mapping cone.  
   May 13: Triangulated categories. 
   May 17: t-Structure and associated abelian category.
   May 19: Perverse sheaves, decomposition theorem, link with D-modules. 

Postscript
   May 22, 2016: the course is now finished and the notes (currently presented as 2 chapters) have approached a more  definitive  form. They reveal that of the course syllabis as announced above, the topic "Quasi-unipotence and  the invariant cycle theorem” has not been reached and that the overview of  "D-modules and mixed Hodge modules was less than scant. This is for some other time perhaps.